United lost 2-0 to Paris Saint-Germain in the UEFA Champions League last week but had previously gone 11 games unbeaten under Solskjaer after he succeeded Jose Mourinho.
There is still no certainty about who will be United's manager in the long term, though, and Tottenham Hotspur's Mauricio Pochettino remains a favoured option, per James Robson in the Evening Standard.
Per Donald McRae of the Guardian, when asked whether he thinks Solskjaer, his former team-mate, should be given the job on a permanent basis, Wales manager Giggs said:
"I do. We've messed around far too long. When you've got someone who knows the club, who is tactically astute, clearly has the players and fans on board, that gives you lots of momentum. There will be bumps in the road but most fans imagine what he could do with the money that's been spent the last few years."
The United legend added that the Red Devils also need to be better with their planning if they are to compete again for the Premier League title:
"I wouldn't be in any rush [to announce Solskjaer as permanent manager] because you don't want to derail the current situation. Planning is more important than the announcement. Look at [Pep] Guardiola. He got in Manchester City players he wanted the year before he took over. So, if it is Ole, keep it amongst the club and start planning. Our problem the past few years is that we were always playing catch-up. We should identify the two or three players we need if we're going to win the Premier League and Champions League."
Giggs has experience of being interim manager at United.
At the end of his remarkable playing career, during which he won the Premier League 13 times with the Red Devils, he joined the Old Trafford coaching staff and helmed the side for four matches at the end of 2013-14 after David Moyes was sacked.
He then worked as assistant manager under Louis van Gaal before departing the club when Mourinho was appointed in 2016.
The 45-year-old long looked the most likely of the numerous stars to play for United under Sir Alex Ferguson to eventually emulate the Scot in becoming Red Devils manager.
Now, though, it is Solskjaer who has brought Ferguson's exciting style of play back to the club.
He will further enhance his credentials to get the job on a permanent basis if he can lead United to victory on Monday against Chelsea in the FA Cup fifth round.
It is a big challenge given United have not won at Stamford Bridge since October 2012, when Ferguson was still in charge.
But Solskjaer's domestic record at United is as close to perfect as anybody could have hoped for, and he will be desperate to see his side bounce back after the PSG defeat.